{
  "abstract": "We answer in the negative the question of Thomas, recorded by Conley, Conley--Marks--Unger, and Conley--Tamuz, of whether every locally finite Borel graph admits a Borel unfriendly partition. Our counterexample has unbounded degree and is closed on a zero-dimensional Polish space; its connectedness relation is hyperfinite, and its components are bipartite and one-ended. Every unfriendly colouring is proper. Together with a parity obstruction, this rigidity rules out Baire measurable colourings that are unfriendly on a comeager set, and measurable colourings that are unfriendly almost everywhere for a quasi-invariant probability of finite average degree. In the positive direction, a Borel graph of maximum degree at most four admits a Borel unfriendly colouring whenever each component contains a cycle or a vertex of degree at most two. This reduces the Borel problem in maximum degree three to cubic forests and, with a theorem of Conley--Marks--Unger, gives Baire measurable unfriendly colourings for all Borel graphs of maximum degree at most four.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11919",
  "author_total": 1,
  "authors": [
    {
      "name": "José de Jesús Pelayo-Gómez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ftkk67papjek.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11919",
  "fulltext_url": null,
  "id": "7ftkk67papjek",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11919",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7212289565",
  "pdf_url": "https://arxiv.org/pdf/2609.11919",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ftkk67papjek.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11919",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11919"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Unfriendly partitions of locally finite Borel graphs",
  "title_text": "Unfriendly partitions of locally finite Borel graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/7ftkk67papjek",
  "venue": "arXiv",
  "versions": []
}